The voltage is not negative, always. The negative sign in Faraday's law (Lenz's law) does not mean that the EMF (or current) always points in some "negative" direction. It means that the current always flows in a way to oppose the change in flux, which is nicely illustrated in that video clip.

Is emf can be negative?

Yes, we can have a neagtive emf. The negative sign implies that the electromotive force induced will be induced in order to oppose the cause. Suppose the emf is induced by a magnetic field increasing in the -z direction, the emf will be induced to produce an magnetic field in the +z direction.

What happens when emf is negative?

The electromotive force (EMF) is the maximum potential difference between two electrodes of a galvanic or voltaic cell. If the cell potential is negative, the reaction is reversed. In this case, the electrode of the galvanic cell should be written in a reversed order.